2.3 | The Derivative of a Function


For any function \(f\), we define the derivative function, \(f’\), by
$$
f'(x) =
\begin{array}{c}
\text{Rate of change} \\
\text{ of } f \text{ at } a
\end{array}
= \lim_{h \rightarrow 0} \frac{f(x+h) -f(x)}{h}.
$$
If \(f'(x) > 0\) on an interval, then \(f\) is increasing on that interval. If \(f'(x) < 0\) on an interval, then \(f\) is decreasing on that interval.
If \(f(x) = k\), then \(f'(x) = 0\).
If \(f(x) = mx + b,\) then \(f'(x) = \text{ Slope } = m.\)
If \(f(x) = x^n,\) then \(f'(x) = n x^{n-1}.\)

E 2.3 Exercises